Bogoliubov Fermi Surface Revealed

Journal Club for Condensed Matter Physics DOI:10.36471/JCCM_October_2020_02 https://www.condmatjclub.org

Bogoliubov Fermi surface revealed

Discovery of segmented Fermi surface induced by Cooper pair momentum Authors: Zhen Zhu, Micha lPapaj, Xiao-Ang Nie, Hao-Ke Xu, Yi-Sheng Gu, Xu Yang, Dandan Guan, Shiyong Wang, Yaoyi Li, Canhua Liu, Jianlin Luo, Zhu-An Xu, Hao Zheng, Liang Fu, and Jin-Feng Jia arXiv:2010.02216

Recommended with a Commentary by Carlo Beenakker, Instituut-Lorentz, Leiden University

The Fermi surface of a metal separates electronic states that are occupied at zero tem- perature from states that remain empty. It is obtained by solving for zero excitation energy, E(p) = 0, to produce a d − 1 dimensional surface in d-dimensional momentum space. Can a superconductor have a Fermi surface, a surface of zero excitation energy for Bogoliubov quasiparticles? That question was addressed theoretically by Volovik many years ago, and an experimental demonstration is now reported by Zhu et al. (arXiv:2010.02216). A figure from Volovik’s 2006 pa- per (arXiv:cond-mat/0601372) explains the mechanism for the emergence of a Bogoli- ubov Fermi surface. The excitation gap ∆ closes if the velocity vs of the Cooper pairs exceeds a critical velocity vc = ∆/pF. This is the Doppler effect first pointed out for su- perfluids by Landau, which shifts the quasiparticle energy by an amount

δE(p) = p · vs. (1) Upon increasing the velocity v of the superconduct- The gap remains open for momentum direc- s ing condensate, the excitation gap closes at a critical tions p⊥ perpendicular to the superflow, pro- velocity v = ∆/p and a Bogoliubov Fermi surface ducing the segmented Fermi surface shown c F appears. The left panel shows the drop in the Cooper in the figure — provided that vs remains be- pair density ρs, which vanishes at the depairing low the depairing velocity v∗ at which the velocity v . The right panel shows the contours of superconductor becomes a normal metal. ∗ zero excitation energy: a Fermi surface formed out of In a bulk superconductor v∗ ≈ vc so electron-like states (red) and hole-like states (blue) ap- Volovik’s mechanism is not operative, but if pears for v < v < v . [Adapted from Volovik (2006).] superconductivity is induced by the proxim- c s ∗ ity effect in a 2D electron gas one may have vc = ∆induced/pF small compared to the depairing velocity in the bulk. This is the approach

1 taken by Zhu et al., following up on a proposal by Yuan and Fu (arXiv:1801.03522). The 2D electron gas on the surface of the topological insulator Bi2Te3 is proximitized by the superconductor NbSe2. An in-plane magnetic field B induces a screening supercurrent over a London penetration depth λ, which boosts the Cooper pair momentum by an amount ps ' eBλ, in-plane and perpendicular to B. The proximity induced gap is reduced be- low the bulk gap, and thus only a small magnetic field is needed to significantly affect the quasiparticle spectrum without strongly impacting the parent superconductor. The 2D electrons on the surface of a 3D topological insulator are massless Dirac fermions, which requires a modification of the usual expression (1) for the Doppler shift of massive Schrödingerelectrons. In the ideal case of an isotropic dispersion, ε(p) = vF|p|, the Doppler shift due to a Cooper pair momentum ps is given by v δE(p) = F p · p . (2) |p| s

For Bi2Te3 the dispersion relation is anisotropic, Heterostructure studied by Zhu et al. The q 2 2 2 2 2 2 Bi2Te3 topological insulator film is suffi- ε(p) = vF |p| + α px(px − 3py) , (3) ciently thin that the NbSe2 superconductor and a more general expression is needed:1 can induce a pair potential in the 2D electron gas on the top surface. An in-plane ∂ε(p) magnetic field B gives a nonzero Cooper δE(p) = · ps. (4) ∂p pair momentum ps ' eBλ. The corresponding critical momentum depends on the direction of the supercurrent, ( ∆/v in the y-direction, p = F (5) c 5 2 4 (∆/vF)(1 − 2 α pF) in the x-direction. To observe the Bogoliubov Fermi surface Zhu et al. use the technique of Fourier- Transform Scanning Tunneling Microscopy (Petersen et al., 1998). Surface electrons scat- tered by defects produce an oscillatory interference pattern in the local density of states (Friedel oscillations), with a spatial periodicity set by the difference of wave vectors on the Fermi surface. The Fourier transform of the spatial dI/dV map measured at low bias voltages with a scanning probe reveals these periodicities.

1 Eq. (4) holds to ﬁrst order in ps for a time-reversally-symmetric single-electron Hamiltonian H0(p) and an s-wave pair potential ∆. The Cooper pair momentum enters in the Bogoliubov-De Gennes Hamiltonian H(p) = H0(p)τz + ∆τx as an oﬀset p 7→ p + τzps, with τz a Pauli matrix in the electron-hole degree of freedom. The linearized energy shift is δE = τzps · h∂H/∂pi = τ0ps · h∂H0/∂pi = τ0ps · ∂hH0i/∂p, in view 2 of Hellmann-Feynman and τz = τ0. Because τ0H0 and H commute, they can be jointly diagonalized and the expectation value hH0i in the basis of eigenstates of H is equal to an eigenvalue ε(p) of H0(p), hence δE = ps · ∂ε/∂p. I have not found this simple formula in the literature and thank Yaroslav Herasymenko for the derivation.

2 Zhu et al. present a detailed numerical analysis of their experimental data, but the key features are evident upon inspection: In the absence of a magnetic field the hexago- nally warped Fermi surface of Eq. (3) is visible in the dI/dV map for voltages outside of the superconducting gap, while at V = 0 the dI/dV map is featureless. Application of an in-plane magnetic field reintroduces the seg- Fourier transformed dI/dV maps, measured at dif- ments of the Fermi surface with wave vectors ferent bias voltages V and magnetic fields B. At in the perpendicular direction. B = 0 the hexagonal Fermi contour of the Bi2Te3 Volovik’s supercurrent mechanism is not surface electrons is visible outside of the supercon- the only way to create an extended Bogoli- ducting gap (e|V | = 2 meV > ∆induced = 0.5 meV), ubov Fermi surface (a gapless d − 1 dimen- while it vanishes inside the gap. An in-plane mag- sional manifold in d-dimensional momentum netic field of 40 mT closes the gap in the perpendic- space). In a multiband superconductor with ular direction. Only the electron-like segments of the a nodal pair potential the pairing of mis- Bogoliubov Fermi surface are probed by the STM for matched Fermi surfaces can expand a nodal V = 0+. point or nodal line into a 2D surface seg- ment. This mechanism could be operative in a dx2−y2 superconductor in a Zeeman field (Yang and Sondhi, 1998) or in a chiral pair potential with intrinsically broken time-reversal symmetry (Agterberg, Brydon, and Timm, 2017; Link and Herbut, 2020). I am not aware of any experimental demonstration along these lines. The simplicity of the realization of Zhu et al. promises much follow-up work. Papaj and Fu (arXiv:2006.06651) propose to confine the gapless superconducting state to a narrow 1D channel, surrounded by 2D regions with a full superconducting gap. Majorana bound states may emerge at the end points of the channel, providing an alternative platform to existing semiconductor nanowire based systems.